Some lower bounds for the numerical radius of Hilbert space operators

We show that if $T$ is a bounded linear operator on a complex Hilbert space, then\begin{equation*}\frac{1}{2}\Vert T\Vert\leq \sqrt{\frac{w^2(T)}{2} + \frac{w(T)}{2}\sqrt{w^2(T) - c^2(T)}} \leq w(T),\end{equation*}where $w(\cdot)$ and $c(\cdot)$ are the numerical radius and the Crawford number, respectively.We then apply it to prove that for each $t\in[0, \frac{1}{2})$ and natural number $k$,\begin{equation*}\frac{(1 + 2t)^{\frac{1}{2k}}}{{2}^{\frac{1}{k}}}m(T)\leq w(T),\end{equation*}where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.